Novel Depth Cues from Uncalibrated Near-field Lighting
Sanjeev J. Koppal and Srinivasa G. Narasimhan
Robotics Institute, Carnegie Mellon University, Pittsburgh, USA
Email: (koppal,srinivas)@ri.cmu.edu
Abstract
Let a near point light source move along a line or in a plane.
Our key idea is to use this light source path as a ’baseline’ with
respect to which scene depth cues can be extracted. Analysis
of the intensities at each pixel yields the following depth cues:
Plane-Scene Intersections: Let a lambertian scene be illuminated by a point source moving along a line. By simply detecting intensity maxima at a scene point, we produce
plane-scene intersections similar to those obtained by sweeping a sheet of light over the scene. Intersections with planes
of other orientations are produced by changing the direction
of the light source path.
Depth Ordering: A point light source moving in a plane
illuminates a homogeneous lambertian scene under orthographic viewing. Under certain conditions, it is possible to
obtain an ordering of the scene in terms of perpendicular distances to this ’base plane’. This is done simply by integrating
the measured intensities at scene points over time.
Mirror Symmetries: An orthographic camera views a
lambertian scene illuminated by a light source moving along
a line. We detect mirror symmetries by reflecting scene points
across a plane containing the light source path. These symmetric pairs will observe identical incident angles and light
source distances and are detected by simply matching scene
points with the same brightnesses over time.
In practice, we relax many of the restricting assumptions
described above. First, we do not require the source to move
precisely along a line or in a plane. Instead, a user can simply
move the source with his/her hand, making image acquisition
easy. Second, simple heuristics can be used to extend all of
our results to non-Lambertian BRDF’s containing diffuse and
specular components. We show strong supporting evidence
using both rendered and real scenes with complex geometry
and material properties.
Our approach computes depth information by moving the
light source, complementing traditional stereo methods that
exploit changes in viewpoint. An interesting similarity to
camera-based methods is that increasing the length of the
source path is similar to increasing stereo baseline and results
in more reliable depth cues. One difference, however, is that
our cues are defined with respect to the light source path. This
allows us to obtain depth cues from different ’viewpoints’,
without moving the camera. These novel scene visualizations
provide strong constraints for any depth estimation technique,
while avoiding the correspondence problem. Our approach
is simple, requiring neither geometric/radiometric calibration
nor a complex setup. Therefore we believe our depth cues
have broad application in computer vision.
We present the first method to compute depth cues from images taken solely under uncalibrated near point lighting. A
stationary scene is illuminated by a point source that is moved
approximately along a line or in a plane. We observe the
brightness profile at each pixel and demonstrate how to obtain
three novel cues: plane-scene intersections, depth ordering
and mirror symmetries. These cues are defined with respect
to the line/plane in which the light source moves, and not the
camera viewpoint. Plane-Scene Intersections are detected by
finding those scene points that are closest to the light source
path at some time instance. Depth Ordering for scenes with
homogeneous BRDFs is obtained by sorting pixels according
to their shortest distances from a plane containing the light
source. Mirror Symmetry pairs for scenes with homogeneous
BRDFs are detected by reflecting scene points across a plane
in which the light source moves. We show analytic results for
Lambertian objects and demonstrate empirical evidence for a
variety of other BRDFs.
1. Distant vs. Near Lighting
Many vision algorithms assume that the scene is illuminated by light sources that are far away, such as the sun or the
sky. In this setting, the incident illumination angles at each
scene point are identical, allowing easier estimation of scene
properties. Approaches that exploit this fact to recover scene
normals and albedos include the classical photometric stereo
technique ([28]) and several extensions for non-lambertian
low parameter BRDFs (Dichromatic ([11]), Diffuse + Specular ([4], [23],[15],[24]), Microfacet ([17],[27])).
Although these algorithms are popular, there are many scenarios where the distant lighting assumption is invalid. These
include indoor, underground and underwater scenes, as well
as outdoor scenes under night lighting. Scene analysis using
near-field lighting is difficult since the incident angles at each
scene point are different. However, the advantage is that the
intensity fall-off in distance from the near light source is encoded in the images. Therefore it becomes possible to extract
both depths and normals of the scene, and many methods have
shown this (Shape from shading ([20], [19], [9]), Photometric stereo ([8], [3], [2], [10]), Calibrated camera-light source
pairs ([15], [7]), Helmholtz stereopsis ([29])). Exploiting the
intensity fall-off requires the 3D position of the source and
therefore most of these techniques assume calibrated lighting.
Instead, if the near light source is uncalibrated, can we still
extract useful information about scene depth?
1
2. Detecting Plane-Scene Intersections
Consider a scene with unknown geometry. Let it be illuminated by an isotropic point light source moving in a straight
line (see Figure 1) at constant speed. At a line position d = i,
the source S is nearest to scene points lying on a plane perpendicular to the light source path. In this section, we show
that maxima will occur in the brightness profiles of all of these
scene points, owing to a minimum in the inverse-square falloff from the point light source. By simply detecting brightness
maxima occurring at every light source position (or frame), we
obtain plane-scene intersections. Our results appear to be created by intersecting the scene with a plane that is translating
along the source path. This is similar in spirit to structured
light striping where a sheet light is swept across the scene.
In the case of structured light striping, brightness maxima are
detected by simply thresholding each image. The difference
here is that we obtain the same effect using an uncalibrated
isotropic near point light source.
Figure 1. Detecting a Plane-Scene Intersection using Brightness
Maxima: Consider a light source moving in a line in front of a scene.
At any line position d = i, the scene points that are closest to the
light source path will display a maxima in their profiles. We use this
maxima to create plane-scene intersections, similar to structured light
results created by intersecting a sheet of light with the scene.
2.1. Maxima in Brightness Profiles
Let us assume, without loss of generality, that the light
source has unit intensity. If the BRDF is given by B, the foreshortening by f , the incident light angles by θs and φs , the
viewing angles given by θv and φv , and the distance between
the light source and the scene point at line position d by R(d),
then the brightness profile E(d) at a scene point is:
E(d) =
B(θs (d), φs (d), θv , φv ).f (d)
F (d)
=
(R(d))2
(R(d))2
(1)
where the F term contains both BRDF and foreshortening.
Taking the derivative with respect to position, d, gives us an
expression for when the maxima of E(d) should occur,
′
′
(R(d))2 F (d) − 2 R (d)R(d) F (d)
E (d) =
=0
(R(d))4
′
(2)
In the next sub-section we will investigate the maxima of
E(d) that occur when the light source is closest to a scene
point. We call these iso-planar maxima since they indicate
scene points that are located on the same plane (Figure 1).
We assume that the scene is Lambertian and that the distance
between the light source path and the scene point is small.
2.2. Plane-Scene Intersections for a Lambertian Scene
Without loss of generality, consider a scene point located at
the origin, P = (0, 0, 0), and let the light source move along
~
a line parallel to the z-axis, and let its 3D position be S(d)
=
(D, 0, d) where D is its closest distance to the origin. The
Figure 2. Plane-scene intersections on a Lambertian Pot: In this
figure we show results of our method applied to a Lambertian clay
pot. We obtain both horizontal and vertical plane intersections by
moving a near point light source as shown by the marked paths.
These results are similar to those obtained from structured light. The
discontinuity in the result for horizontal planes is due to merging the
intersections for the right and left halves of the pot. These were done
separately to avoid blocking the camera view. Please see [13] for
better visualization.
distance from the scene point to the light source is R(d) =
(D2 + d2 )0.5 . For a scene point with albedo ρ and surface
normal ~n = (nx , ny , nz ) we write:
F (d) =
~
ρ ~n.(S(d)
− P~ )
R(d)
(3)
′
An iso-planar maxima occurs if F (d) is zero when the
light source reaches the closest distance to the scene point.
′
′
Consider the expressions for F (d) and R (d),
′
′
F (d) =
ρR(d).(nz ) − ρR (d)(Dnx + nz d)
(R(d))2
(4)
d
R(d)
(5)
′
R (d) =
′
Putting these into E (d),
′
E (d) =
−ρ(2nz d2 + 3Dnx d − D2 nz )
(R(d))5
(6)
′
Setting E (d) = 0 gives us a quadratic equation in d with
two possible solutions. By checking the sign of the second
′′
derivative, E (d), for all possible normals, we found that one
of these solutions is always a maxima and is given by,

dmaxima = D 
− 43 +
q
( 34 )2 + 21 ( nnxz )2
nz
nx


(7)
Ideally, iso-planar maxima should occur when d = 0, since
that is when the light source is closest to the scene point and
R(0) = D. Therefore, Equation 7 represents the error in
the iso-planar maxima location. However, simulations show
that this error remains bounded as nnxz is varied. For instance, dmaxima is zero when nnxz → 0 and becomes √D2 when
nz
nx → ∞. Although we will investigate this bound more thoroughly in future work, here we assume the error is negligible
if the light source path is close to the scene point (D is small).
In Figure 2 we show both horizontal and vertical planescene intersections for a lambertian pot, created by moving
the light source first sideways and then upwards. We code the
planes from blue to red, obtaining a continuum of color coded
plane-scene intersections. For the second result, we merged
two experiments for the left and right halves of the pot, to
avoid blocking camera’s view of the scene. Although these
results appear similar to structured light images obtained by
sweeping a plane over the scene, they were obtained by a user
hand-waving a near point light.
Shadows and Specularities: Non-isoplanar brightness
maxima are usually rare in lambertian scenes illuminated by
a near light source. In fact, for a maxima to occur when
′
′
R (d) 6= 0, a fortuitous occurrence of values for R (d), R(d),
′
F (d) and F (d) would be required in Equation 2, which is
less likely. However, scenes with sharp specularities still show
non-isoplanar maxima. Fortunately, since specularities are
characterized by a rapid rise in brightness, these are detected
by thresholding the second derivative of the measured intensities. Glossy highlights are similarly removed using a method
proposed by [16]. Shadows can eliminate iso-planar maxima
from a profile, but are handled by repeating the experiment
with different parallel light source paths. Since a scene point
shadowed in one experiment may be illuminated in another,
we usually detect an iso-planar maxima. Finally, many scenes
may still exhibit non-isoplanar maxima due to material properties. We remove these by simply enforcing neighboring scene
points to have iso-planar maxima that occur closely in time.
Figure 3. Plane-Scene Intersections for Rendered Scenes: We used
a ray tracing software ([18]) to render two objects (dragon and bunny)
using both the Lambertian + Torrance-Sparrow and the Oren-Nayar
models. For each set of images, we created plane-scene intersections,
and we show two examples of these at the top of the figure. We fit a
plane to each intersected region and measure the plane fit error. The
average of these errors is plotted, as a percent of the longest distance
contained in the object. These empirical results support the idea that
the plane-intersection algorithm can be used with a variety of nonlambertian scenes.
2.3. Experimental Results
It is possible to create iso-planar maxima in profiles if the
′
derivative of the BRDF, F (d), is zero around a small interval
′
where R (d) = 0. In other words, the change in BRDF must
be negligible for a short interval around the maxima location.
This seems a reasonable assumption for diffuse BRDFs. In
Figure 3, we show empirical evidence supporting this idea
for scenes created using a ray-tracing tool ([18]). We rendered the bunny and dragon models by varying the parameters of the Oren-Nayar model (facet angle from 0 to π2 ) and
of a Lambertian + Torrance-Sparrow model (σ of facet distribution from 0 to 1). The light source was moved along the
Figure 5. Overlap of incident angles for a simple 2D scene: A light
source moving in a line illuminates a 2D scene with homogeneous
BRDF. Let scene point P be at a greater depth than Q and let both
have the same surface normal. Some of the incident angles at P are
repeated in Q, and we show these overlapped angles in gray. Because
of inverse-square fall-off, the measured intensities in the overlapped
region will be higher in Q than in P. We extend this idea to create
scene depth ordering for 3D scenes as well.
3. Depth Ordering for Homogeneous Scenes
Figure 4. Plane-Scene Intersections for Real Scenes: At the top
of the figure we show horizontal and vertical plane-scene intersections for a painted house, even though this scene demonstrates glossy
specularities. We also show plane-scene intersections for an office
desk made of metal and plastic. At the bottom we show the subsurface scattering effects of a wax candle, by shining a point laser.
Our method is able to create horizontal plane-scene intersections for
this object with complex BSSRDF ([6]) appearance.
x− and then y− axes to create horizontal and vertical planescene intersections. Using ground-truth, we fit a plane to the
3D locations of scene points in the plane-scene intersections
and plotted the sum-of-squared errors. The low errors indicate
the robustness of our technique for non-lambertian BRDFs.
At the top of Figure 4, we show horizontal and vertical
plane-scene intersections for a painted house model, which
are detected despite glossy specularities in the scene. Our
algorithm also produces good results for scenes with sharp
specularities, such as the metal office desk in Figure 4 and
for rough objects with cracks, such as an earthen pot in Figure
4 (see [13] for the input sequence and a better visualization
of the plane-scene intersections). We are also able to create
plane-scene intersections for objects with sub-surface scattering (since BSSRDF ([6]) is smooth) such as a wax candle
shown in Figure 4. The scattering effects are demonstrated
using a laser pointer, and horizontal plane-scene intersections
are shown. Such a result would be hard to obtain using traditional structured light methods.
Consider a scene with homogeneous BRDF viewed by an
orthographic camera. This scene is illuminated by a point
source moving in a plane. Intuitively, scene points closer to
the light source tend to be brighter than scene points further
away. If we somehow manage to remove the effect of BRDF
for any two scene points, then their appearance would depend only on their distances to the light source. Although this
is not possible generally, we describe certain conditions under which we obtain scene depth ordering with respect to the
”base plane” containing the light source path. We will provide two heuristics to achieve good results for depth ordering:
a) Move the light source over a large area of the base plane
and b) Bring the base plane as close as possible to the scene.
We support our method with strong empirical evidence using
both simulations and real scenes.
3.1. Integrating the Brightness Profile
Consider a scene point P illuminated by a light source
moving in a plane. As before, we will assume the light source
has unit intensity and let F contain both the BRDF and foreshortening. If R(d) is the distance between the source and
this scene point at position d on the light source path, then the
measured intensity profile of P is given as:
Ep (d) =
Fp (d)
.
(Rp (d))2
(8)
Let Sp be the sum of the P ’s intensities along the light
source path from positions d1 to dn ,
Sp =
dn
X
d=d1
Fp (d)
.
(Rp (d))2
(9)
Let there be another point Q whose perpendicular distance
to the plane containing the light source is less than that of P .
Figure 6. Light Source Moves in a Path with Finite Length: A
light source moving in a line illuminates a scene with homogeneous
BRDF. Our depth ordering method works for scenes where any point
at a greater depth than Q must be further away from the light source
at every time instance. This restricts us to a non-planar region around
Q as shown.
At different light source positions, P and Q may observe identical incident angles. We term these as overlapped incident
angles, and they are shown in gray in Figure 5. In this simple
2D scene, the longer the length of the light source path, the
greater the overlapped region. For an infinite path, all the incident angles at P and Q will overlap. Since the inverse-square
fall-off is smaller at Q than at P , Sp < Sq .
This result is harder to demonstrate for real 3D scenes
where P and Q can have different surface normals. To make
the analysis easier, we assume that the light source is always
further from P than Q. This assumption of Rp (d) > Rq (d)
for all positions along the light source path is illustrated in
Figure 6. We will show that this drawback is not severe since,
in practice, we get good depth ordering results for scenes with
a variety of geometries.
Sp and Sq consist of two components, one of which (denoted by O) comes from overlapped incident angles, as in the
gray region in Figure 5. The other comes from non-overlapped
or different incident angles (denoted by N ). Since the order of
the summation does not matter, we let the overlapped angles
occur from d1 to di , and the non-overlapped from di+1 to dn .
We separate the summation accordingly as:
Sp =
di
X
d=d1
dn
X
Fp (d)
Fp (d)
+
2
(Rp (d))
(Rp (d))2
(10)
d=di+1
which we write concisely as:
Sp = Op + Np
(11)
Similarly for scene point Q we have Sq = Oq + Nq . The
overlapped terms, Op and Oq , have the same incident angles. Since P is further away than Q, the inverse-square falloff from the light source ensures that Op < Oq . Therefore,
Sp < Sq if Np < Nq . Consider a pair of the summation terms
from Np and Nq . If the inequality holds for each of these
pairs, then,
Figure 7. Simulations of convex and concave Lambertian objects:
Our depth ordering method always works for convex lambertian objects. We show results with very low errors on renderings of a convex
shape whose curvature we gradually increase. We also render concave objects at different curvatures. Although the depth orderings are
much worse in this case, they stabilize and do not degrade with time.
In addition, bringing the light source closer creates better orderings
for shallow concave objects.
Fq (d)
Fp (d)
≤
2
(Rp (d))
(Rq (d))2
(12)
3.2. Depth Ordering for Lambertian Scenes
When does Equation 12 hold? We answer this question by
first investigating depth ordering for lambertian scenes. We
divide the problem into two cases, based on whether the local geometry between the scene points P and Q is convex or
concave.
Convex neighborhood: If the normals at P and Q are
given by np and nq and if the 3D position of the light source
is s(d) then we can rewrite Equation 12 as,
np .(s(d) − P)
≤
nq .(s(d) − Q)
Rp (d)
Rq (d)
3
(13)
For a convex object, a scene point’s foreshortening to the
light source is higher than one farther away. Therefore the
LHS of Equation 13 is always less than one, and our method
always works for pairs in a convex lambertian neighborhood.
In Figure 7 we show simulations of depth orderings for convex objects of different curvatures demonstrating above 97%
accuracy. Small errors occur due to violations of the assumption in Figure 6. We also present results on non-lambertian
convex objects, such as corduroy and dull plastic (Figure 9).
Figure 9. Depth Ordering Results on Real Convex Scenes: We
acquired images by waving a near point light in a plane. For each
scenes we display the depth ordering obtained as a ”pseudo depth”,
by plotting the ordering in 3D. The objects presented are made up of
a variety of materials such as plastic and corduroy.
Figure 8. Depth Ordering for Rendered Scenes: We used the PBRT
software ([18]) to render three scenes using both the Lambertian +
Torrance-Sparrow and Oren-Nayar models. We covered the space of
roughness parameters for each of these models. Using ground truth
depths for these scenes, we measured the percentage of correctly ordered scene points and we plot the results above. These results provide empirical evidence supporting the use of our method with diffuse non-lambertian BRDFs.
Concave neighborhood (shadows and interreflections):
The foreshortening of P and Q are now opposite to the convex
case, and the LHS of Equation 13 is always greater than 1. In
fact, the further the two points are, the larger the left hand side
of the equation will be. Fortunately, our method still performs
well thanks to the effect of shadows. We rewrite Equation 13
by adding a visibility term, V (d),
V (d)
np .(s(d) − P)
≤
nq .(s(d) − Q)
Rp (d)
Rq (d)
3
(14)
Since P is further away from the light source than Q, it is
more often in shadow. Whenever this happens, V (d) = 0 and
Equation 14 is true. Concave depth ordering has more errors
than the convex case (Figure 7), the results are still reasonable (above 82%). In addition, the closer we bring the light
source to the scene, the better the results get since the RHS
of Equation 14 increases. Finally, we note that smooth interreflections do not severely degrade performance, since the errors for concave depth ordering remain roughly constant with
increase in curvature.
3.3. Experimental Results
In Figure 8, we have rendered three scenes (buddha, bunny
and dragon), using ray tracing ([18]). As before, we varied the
roughness parameters for the Oren-Nayar model (facet angle
from 0 to π2 ) and the Lambertian + Torrance-Sparrow model
(σ of facet distribution from 0 to 50), getting 10 sets of images,
each with 170 different light source locations. We compare
the depth ordering with ground truth depth and show accuracy
above 85% for the Oren-Nayar model and above 75% for the
Lambertian + Torrance-Sparrow model. In Figure 10 we display the ordering for a polyresin statue containing shadows
and interreflections as a ”pseudo depth” in Maya. We have
shown different views as well as its appearance under novel
lighting. Finally, an advantage of our method is that these
depth orderings are with respect to the light source and not
the camera. We obtain orderings from different planes, without camera motion and avoiding the difficult problem of scene
point correspondence. In Figure 11 we show ordering results
for a wooden chair, for three different planes.
Figure 10. Depth Ordering Results for a Statue: In the top row we
show a polyresin statue and a 2D depth ordering image obtained from
an uncalibrated near light source moving in a plane. We visualize the
ordering as a ”pseudo depth” in the bottom row, applying novel lighting, viewpoint and BRDF. Although this object has complex effects
such as interreflections and self-shadowing, our depth ordering is accurate, as we show in the close-up. Please see [13] for many other
view points.
Figure 11. Depth Ordering from different viewpoints: Here we
show depth ordering of a wooden chair from the viewpoints of three
different planes. Brighter scene points are closer than farther ones.
The second depth ordering is from the camera point of view, and is
similar to conventional depth results obtained from stereo algorithms.
We also get the orderings from the left and right planes without moving the camera. This allows us to obtain novel visualizations of the
scene, while avoiding the problem of scene point correspondence.
4. Mirror Symmetries in Homogeneous Scenes
be sparse, since the reflection plane would not coincide with
the mirror symmetry of the object.
In this section, we will investigate a way of finding mirror symmetry pairs for scenes with homogenous BRDFs. Although our method is simple, it is useful for objects that have
the mirror symmetry in their shapes and produces very sparse
pairs otherwise.
Mirror Symmetry Pairs in Lambertian Scenes Consider
a point P in a homogeneous Lambertian scene containing a
mirror symmetry, viewed by an orthographic camera. Let the
scene be illuminated by a point light moving in a plane parallel
to the viewing direction. Reflecting P and its surface normal
across such a plane does not change its incident elevation angles and light source distances (Figure 12). Therefore, all the
reflected pairs will have identical appearance over time and
can be easily located. We show the result of matching identical profiles for a homogeneous Lambertian pot at the top of
Figure 13. The pot has a reflection symmetry across a vertical
plane passing through its center and perpendicular to the image plane. We wave a point light source in this plane and mark
the matched scene points with the same color. Note that if the
light source was moved in a different plane, different symmetry pairs would be obtained. However, these would most likely
Shadows and Inter-reflections: A scene with mirror symmetry has the same geometry across the plane of reflection.
Therefore, geometric appearance effects such as shadows and
inter-reflections will be identical between symmetry pairs, creating identical intensity profiles.
Isotropic BRDFs: In homogeneous non-lambertian scenes
the effect of the incident azimuth angles must be considered.
In general these will be different for P and its symmetry pair.
However, the absolute difference between the azimuth angles
are the same, and therefore our method works for isotropic
BRDFs. As before, the symmetry pair of P is found by reflection across the plane containing the light source path and
the orthographic viewing direction. Using the laws of symmetry we can easily show that P and its symmetric pair have the
same light source distances, the same elevation angles, and
identical absolute differences between the azimuth angles of
viewing and illumination. We find symmetry pairs by matching identical intensities for an office desk at the bottom of Figure 13, even though this scene contains non-lambertian materials such as metal and plastic.
6. Acknowledgements
This research was partly supported by NSF awards #IIS0643628 and #CCF-0541307, and ONR contracts #N0001405-1-0188 and #N00014-06-1-0762. The authors thank Mohit
Gupta and Shuntaro Yamazaki for useful technical discussions.
References
Figure 12. Matching Identical Profiles to give Symmetry Pairs:
We show a light source moving in a line illuminating a scene point P.
We find a symmetry pair by reflecting P across a plane containing the
light source path and the orthographic viewing vector. The elevation
angles (θ) and the absolute differences in azimuth angles (φ) are the
same for the pair.
Figure 13. Symmetry Pairs in Real Scenes: We show two real
scenes, a lambertian pot and a office desk. A near point light source
was hand-waved approximately along the axis of reflection. We
match identical profiles in the scene and mark them with the same
color, giving us symmetric pairs in the scene.
5. Conclusions
Algorithms for obtaining scene geometric information
from uncalibrated, varying lighting are well known in the case
of distant light sources. For example, many photometric stereo
methods have been introduced that handle unknown, distant illumination ([5], [12], [1], [26]). In addition, variation in shadows has been used extensively ([22], [14], [21]) to obtain cues
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distant illumination scenario. This is surprising, since, intuitively, the more complex the set of rays that illuminate the
scene, the richer should be the information obtained from that
scene. In this paper, we have made a first step towards understanding this concept for near point light sources.
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